A characterization of residually finite HNN-extensions of finitely generated abelian groups

Subgroup Separability of HNN-Extensions with Abelian Base Group

“…Assume that G is subgroup separable. Then G is residually finite and by [4] either A = K or B = K or there exist a finitely generated normal subgroup H of G such that H is a subgroup of finite index in D where D is the group D = x ∈ K for every n ∈ there exists a λ n ∈ such that t −n x λ n t n ∈ K If A = K B = K or B = K A = K we get a contradiction by using Lemma 1. On the other hand, if the subgroup H described above exists then in view of Lemma 2 it suffices to show that A A ∩ B < ∞ and B A ∩ B < ∞.…”

The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: a characterization

“…In the present paper we generalize the results of [1] to the case where the base group is a finitely generated nilpotent group. M. Shirvani, in [8] and [9], gives a necessary condition for an amalgamated free product or an //AW-extension to be residually finite in the case where the factors of the amalgamated free product or the base group of the //AW-extension satisfy a non-trivial law.…”

“…In [1], in the case where, in the //TVJV-extension G -(t, K\t~lAt = B, <p), [2] Residual finiteness of HNN-extemions 409 the base group AT is a finitely generated abelian group, a simple necessary and sufficient condition for the group G to be residually finite was given. In the present paper we generalize the results of [1] to the case where the base group is a finitely generated nilpotent group.…”

Subgroup separability of graphs of abelian groups

“…This last category of groups was completely characterised in [3]. Furthermore, the non-subgroup separable example of Mihailova in [13] contains a subgroup isomorphic to L (see Corollary 1), and the examples of Gitik and Rips in [9] and of Allenby and Doniz in [1] are special cases of our main theorem (see Corollaries 3 and 2).…”